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Metamath Proof Explorer

Theorem itg2cl

Description: The integral of a nonnegative real function is an extended real number. (Contributed by Mario Carneiro, 28-Jun-2014)

Ref Expression
Assertion itg2cl ( 𝐹 : ℝ ⟶ ( 0 [,] +∞ ) → ( ∫2 ‘ 𝐹 ) ∈ ℝ* )

Proof

Step Hyp Ref Expression
1 eqid ⊢ { 𝑥 ∣ ∃ 𝑔 ∈ dom ∫1 ( 𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = ( ∫1 ‘ 𝑔 ) ) } = { 𝑥 ∣ ∃ 𝑔 ∈ dom ∫1 ( 𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = ( ∫1 ‘ 𝑔 ) ) }
2 1 itg2val ⊢ ( 𝐹 : ℝ ⟶ ( 0 [,] +∞ ) → ( ∫2 ‘ 𝐹 ) = sup ( { 𝑥 ∣ ∃ 𝑔 ∈ dom ∫1 ( 𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = ( ∫1 ‘ 𝑔 ) ) } , ℝ* , < ) )
3 1 itg2lcl ⊢ { 𝑥 ∣ ∃ 𝑔 ∈ dom ∫1 ( 𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = ( ∫1 ‘ 𝑔 ) ) } ⊆ ℝ*
4 supxrcl ⊢ ( { 𝑥 ∣ ∃ 𝑔 ∈ dom ∫1 ( 𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = ( ∫1 ‘ 𝑔 ) ) } ⊆ ℝ* → sup ( { 𝑥 ∣ ∃ 𝑔 ∈ dom ∫1 ( 𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = ( ∫1 ‘ 𝑔 ) ) } , ℝ* , < ) ∈ ℝ* )
5 3 4 ax-mp ⊢ sup ( { 𝑥 ∣ ∃ 𝑔 ∈ dom ∫1 ( 𝑔 ∘r ≤ 𝐹 ∧ 𝑥 = ( ∫1 ‘ 𝑔 ) ) } , ℝ* , < ) ∈ ℝ*
6 2 5 eqeltrdi ⊢ ( 𝐹 : ℝ ⟶ ( 0 [,] +∞ ) → ( ∫2 ‘ 𝐹 ) ∈ ℝ* )